3 HAUSDORFF SPACES 9 (c) In a topological space E, a subset F is irreducible if and only if its closure F is irreducible. In particular, for each x e E, the set {x} is irreducible. If an irreducible set F in E is of the form {x}, then x is said to be a generic point of F. (d) Let A be an integral domain. Show that the topological space Spec(A) (Problem 4) is irreducible and that {0} is its unique generic point. In Spec(Z), show that the complement of the generic point is an irreducible set which has no generic point. (e) If E is an irreducible space, every nonempty open set in E is irreducible. (f) Let (U«)a6 A be an open covering of a topological space E, such that U« n U/» =£ 0 for all pairs of indices (a, ft). Show that if the sets U« are irreducible, then E is irreducible. (g) Let E, F be two topological spaces and let/be a continuous mapping of E into F. If A is an irreducible subset of E, show that/(A) is an irreducible subset of F. (h) Let E, F be two irreducible spaces, each of which has at least one generic point. Suppose also that F has a unique generic point b. Let/be a continuous mapping of E into F. Show that/(E) is dense in F if and only iff(x) = b for each generic point x of E. 6. A topological space E is said to be quasi-compact if it satisfies the Borel-Lebesgue axiom (3.16) and compact if it is quasi-compact and Hausdorff. A subset F of a topo- logical space E is said to be a quasi-compact set (resp. a compact set) if the subspace F of E is quasi-compact (resp. compact). Every finite set is quasi-compact. (a) In a quasi-compact space, every closed set is quasi-compact. (b) Let E be a Hausdorff space and let A, B be two disjoint compact sets in E. Show that there exist two disjoint open sets U, V, such that A c U and B cr V. (Consider first the case where A consists of a single point.) Deduce that a compact set in a Hausdorff space is closed. (c) In a topological space E, any finite union of quasi-compact sets is quasi-compact. (d) If E is a quasi-compact space and /: E -* F is a continuous mapping, then the set /(E) is quasi-compact. (e) A filter base on a set E is a set 93 c $(E) of nonempty subsets of E such that, whenever X and Y belong to 93, there exists Z e 93 such that Z <= X n Y. Show that if E is a quasi-compact space and all the sets of 93 are closed, then the intersection of the sets of 93 is nonempty (consider the complements of the sets of 93, and argue by con- tradiction). (f) If A is a commutative ring with an identity element 1^0, show that the topological space Spec(A) (Problem 4) is quasi-compact. (g) Let E be the union of the interval ]0, 1 ] and two elements a, ft. Let 93 be the set of finite intersections of sets of the forms ]a, !],{«} u ]0, a[,{ft} u ]0, a[, where 0<a<l. Show that 93 is a basis of a non-Hausdorff topology on E, for which E is quasi-compact and every set consisting of a single point is closed in E. Every point of E has a compact metrizable neighborhood, but there are such neighborhoods of a (or ft) which are not closed in E. Show that the intersection of a compact metrizable neighborhood of a and a compact metrizable neighborhood of ft is not quasi-compact. The topology of E has a denumerable basis consisting of separable metrizable subspaces. (h) Let E be the sum (1.8) of N and an infinite set A. For each xeN, write @(#) = {x}. For each x G A, let S*, „ denote the union of {x} and the set of integers J>/i, and let <&(x) denote the set of sets S*, „ as n runs through N. Show that there exists a non-Hausdorff topology on E such that @(x) is a fundamental system of neighborhoods of x, for each x e E. In this topology, every set consisting of a single point is closed, and E has a dense compact subspace, although E itself is not quasi-compact.